Semiorthogonal Decompositions on Total Spaces of Tautological Bundles

Abstract: Let U be the tautological subbundle on the Grassmannian

m a t h r m G r (k, n)

. There is a natural morphism

m a t h r m T o t (U) o m a t h b b A^{n}

. Using it, we give a semiorthogonal decomposition for the bounded derived category

D_{m a t h r m c o h}^{b} (m a t h r m T o t (U))

into several exceptional objects and several copies of

D_{m a t h r m c o h}^{b} (m a t h b b A^{n})

. We also prove a global version of this result: given a vector bundle

E

with a regular section

s

, consider a subvariety of the relative Grassmannian

m a t h r m G r (k, E)

of those subspaces which contain the value of

s

. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of

s

. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case

k = 1

.

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